The 'top of atmosphere' theory of global warming is based on two actual facts:
a) The gravito-thermal effect exists (as adjusted for latent heat of evaporation/condensation), which is why temperatures fall by ~6.6 degrees/km as you go up through the troposphere (agreed).
b) Atmospheric density falls as you go up; so the amount of CO2 per m3 falls as you go up (agreed).
The theory says that a significant amount of radiation emitted below the 'effective emitting altitude' (at which CO2/m3 is a certain critical value) cannot escape to space. Only as much radiation makes it past the 'effective emitting altitude' as a planet receives from the sun, so they balance out and surface temperatures stabilise. If overall CO2 levels increase, the effective emitting altitude goes up. Therefore, surface temperatures must go up - you are adding more km x 6.6 degrees/km to a constant to work back down to the surface temperature.
The theory is intellectually appealing and it's difficult to prove or disprove the logic just using logic. All we can do is test the theory against 'real life'.
In the first test, we established that CO2/m3 at Earth's effective emitting altitude (~ 5km up) is ~0.5g/m3. If the theory holds, the effective emitting altitude pre-1880 would have been ~3km up ≈2 km lower than now and surface temperatures would have been ~13 degrees lower than now. Surface temperatures (at the end of the Little Ice Age, natch) were only ~1.3 degrees lower than now, so the theory looks flawed to say the least.
I have given this some more thought and there's another way of testing the theory; we can apply the same logic to Venus, Earth and Mars and see if CO2/m3 levels at each planet's effective emitting altitude are similar. If they are, then that backs up the theory.
We find that the CO2/m3 level at the effective emitting altitude is ~3,700 g/m3 on Venus; ~0.5 g/m3 on Earth and ~17 g/m3 on Mars. The calculations are tortuous (took me all yesterday evening) so I won't bore you with them and there are large margins of error. The three planet's effective temperatures are quite similar (~230K, ~255K and ~205K) so I see no need to adjust for that. These CO2/m3 figures are so far out of line as to be meaningless. So the theory fails this test even worse that it failed the first test.